On the inverse problem of the product of a form by a monomial: the case \(n=4\). I. (English) Zbl 1184.42022
In this paper, the authors consider the following inverse problem for linear functionals. Given a regular linear functional \(v\), find all the regular linear functionals \(u\) such that they satisfy the equation \(x^4 u= -\lambda v\), where \(\lambda\) is a complex number different from zero.
First they obtain the regularity conditions and the coefficients of the second order recurrence relation satisfied by the monic orthogonal polynomial sequence with respect to the functional \(u\). They also prove that if \(v\) is semi-classical then \(u\) is semi-classical and they give some results concerned with the class of \(u\). When the form \(v\) is symmetric positive definite, then simpler regularity conditions are given.
Finally, they present an example with the regular form \(u\) semi-classical of class \(4\) and they obtain its integral representation.
First they obtain the regularity conditions and the coefficients of the second order recurrence relation satisfied by the monic orthogonal polynomial sequence with respect to the functional \(u\). They also prove that if \(v\) is semi-classical then \(u\) is semi-classical and they give some results concerned with the class of \(u\). When the form \(v\) is symmetric positive definite, then simpler regularity conditions are given.
Finally, they present an example with the regular form \(u\) semi-classical of class \(4\) and they obtain its integral representation.
Reviewer: Alicia Cachafeiro López (Vigo)
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
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